How to Find Domain and Range of Trigonometric Functions?

The domain and range of trigonometric functions can be found by examining where the function is defined and the function’s output values for each input value.

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A step-by-step guide to the domain and range of trigonometric functions

The domain and range of trigonometric functions are the input values and the output values of trigonometric functions, respectively. The domain of trigonometric functions shows the values of the angles in which the trigonometric functions are defined. The range of trigonometric functions shows the value of the result of the trigonometric function corresponding to a specific angle in the domain.

 Sine function:

We know that the sine function is the ratio of the perpendicular and hypotenuse of a right-angled triangle. The domain and range of trigonometric function sine are obtained as follows:

• Domain \(=\) \(\color{blue}{All\:real\:numbers,\:i.e.,\:(−∞,\:∞)}\)
• Range \(=\color{blue}{ [-1, 1]}\)

Cosine function:

The cosine function is the ratio of the adjacent side and the hypotenuse of a right triangle. The domain and range of trigonometric function cosine are obtained as follows:

• Domain \(=\) \(\color{blue}{All\:real\:numbers,\:i.e.,\:(−∞,\:∞)}\)
• Range \(=\color{blue}{ [-1, 1]}\)

Tangent function:

The tangent function can be written as the ratio of the sine and cosine functions, so the domain of \(tan x\) does not contain values where \(cos x\) is zero. We know that \(cos x\) is \(0\) in odd integral multiples of \(\frac{π}{2}\), so the domain and range of trigonometric function tangent are obtained as follows:

• Domain \(=\) \(\color{blue}{R – (2n + 1)\frac{π}{2}}\)
• Range \(=\) \(\color{blue}{(−∞, ∞)}\)

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Cotangent function:

The cotangent function can be written as the ratio of the cosine and the sine function, and \(cot x\) is the reciprocal of \(tan x\). Therefore, the range of \(cot x\) does not contain values where \(sin x\) is equal to zero. We know that \(sin x\) in integral multiples of \(π\) is \(0\), so the domain and range of trigonometric function cotangent are given by:

• Domain \(=\)\(\color{blue}{R – nπ}\)
• Range \(=\) \(\color{blue}{(−∞, ∞)}\)

Secant function:

It can also be written as the reciprocal of the cosine function. Therefore, the domain of sec x does not contain values where \(cos x\) is equal to zero. \(cos x\) is \(0\) in odd integral multiples of \(π\), so the domain and range of trigonometric function secant are obtained as follows:

• Domain \(=\) \(\color{blue}{R – (2n + 1)\frac{π}{2}}\)
• Range \(=\) \(\color{blue}{(-∞, -1] \cup [+1, +∞)}\)

Cosecant function:

The cosecant function can also be written as the reciprocal of the sine function. Therefore the domain of trigonometric function \(cosec x\) does not contain values where \(sin x\) is equal to zero. We know that \(sin x\) is \(0\) at integral multiples of \(π\), hence the domain and range of trigonometric function cosecant are given by:

• Domain \(=\) \(\color{blue}{R – nπ}\)
• Range \(=\) \(\color{blue}{(-∞,\:-1]\:\cup \:[+1,\:+∞)}\)

Domain and Range of Trigonometric Functions, Example 1:

Find the domain and range of \(y= cos(x)\: – 3\)

Solution:

• Domain: \(\:\left(-\infty \:,\:\infty \:\right)\)
• Range: \(-1\le \:cos\:x\le \:1\:⇒-1\:-\:3\le \:cos\:x-3\le \:1-3\:⇒\:-4\le \:y\le -2\:\)